How do you make mini-planet photos like this one?
By Guido Radig (Own work) [CC BY 3.0 (http://creativecommons.org/licenses/by/3.0)], via Wikimedia Commons
Here are some examples of what I mean on Flickr: Making Planets - though I would be open to a better source of examples.
Answer
The basics is that you start with a 360-degree panorama and apply a rectangular to polar transformation.
I happen to have written introductory tutorials for both:
One easy short-cut is that some recent Fuji cameras actually can produce a seamless 360° panorama right in the camera. Make sure you choose the Cylindrical 360° option under the Motion Panorama function. Otherwise, or if your camera does not have it, then a seam will appear in the results.
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